# What's the difference between magnetic flux density,B and magnectic field density,H?

1. Oct 30, 2004

### tmwong

i know that B = μH, where B is the magnetic flux density, H is magnetic field density and μ is the permeability of the medium. however, i'm wondering what's the difference between magnetic flux density,B and magnectic field density,H?

besides, i have the same problem about an electrostatic equation, which is
D = εE , where D is electric flux density, E is electric field density and ε is permitivity of the medium. why there's still having D while we already have E, since both of the D and E have the same unit.
is their difference is that the magnitud of B and D are dependent to the permitivity and permeability of the medium respectively? on the other hand, H and E are independent of the medium?

2. Oct 30, 2004

### Clausius2

No it isn't. H and D are independent of the medium. E and B are dependent on the medium.

3. Nov 2, 2004

### tmwong

yes. u r right.after a deeper thinking, i'd found a conclusion about their differences. B and D are independent of the medium while E and H are not. that's because H and D are fluks densities, and according to fluks conservative law, the density of fluks(or the magnitude of total enclosed fluks) r always same and cannot change(decrease or increase in magnitude) in the same enclosed surface. however, the magnitud of field densities, E and B are dependent to the permeability and permitivity of te medium respectively.

4. Mar 17, 2011

### audioresearch

Re: what's the difference between magnetic flux density,B and magnectic field density

I am also confused by B and H.

In some texts, they show B pointing one direction inside a magnet but H pointing the opposite direction.

If B = uH, then how can it ever be true that the B vector can point in any direction other than the direction of the H vector inside a material?

What follows are my beliefs about B & H, please critique them, thanks!!

I consider that B is the "real", total field, the one that would deflect a test charge traveling through it, as F is proportional to B cross V (velocity of the moving electric charge). If one had a magnetic field measuring device that could be inserted inside a material having some nonzero value of u, the only thing it could measure there would be the magnitude of the B vector and possibly also the direction of the B vector. Inside that material, it would not be possible to measure H because there is no way to separate the magnetization vector M from the H vector and only their sum, B, would be measurable.

I believe that H is the field in a vacuum that is produced by, say, a current carrying solenoid.

I believe that if you then move into that H field a material that has magnetic domains that can be "swiveled around" by that H field, those domains produce their own magnetic field vector M.

I believe that M vector adds to the H vector and their sum is the B vector.

If I'm wrong about any of this, please try to figure out the error of my thinking and point it out to me.

Thank you very much.

5. Mar 17, 2011

### cabraham

Re: what's the difference between magnetic flux density,B and magnectic field density

Which one is independent of medium varies. It can be E or D for electric, & it can be B or H for magnetic. Take a parallel plate capacitor with 2 dielectrics in parallel, air & plastic. The plastic has a relative permittivity of 10, eps_r = 10. The boundary conditions are that for the 2 media, their tangential components of E must be equal. Hence, Eair = Eplastic. But the D values depend on the medium. In this case, Dplastic = eps_r*Dair = 10*Dair. Of course, D = eps*E = eps_r*eps0*E. In this case E is independent of the medium, where D scales according to medium's permittivity.

But if the dielectrics are stacked in series, we have a different relation. The normal components of D must either be equal, or can at most differ by only a constant. So we have: Dplastic - Dair = rho_s, where rho_s is equal to the surface charge per unit area. If there is no surface charge, then Dair = Dplastic. But the E components scale according to the dielectric constants of the 2 media. Eair = 10*Eplastic. In this case D is independent of the medium, where E scales according to the inverse of the medium's permittivity.

In the magnetic domain, we have a similar relation. Take a magnetic core, excited by a current in its winding. The core has an air gap. Let's say the core is ferrite w/ a relative permeability of 100, & the gap is air (mu_r = 1). The boundary condition here is that the B field normal components are the same for both media. So, Bair = Bfe. But the H values scale in inverse proportion to the permeability, i.e. Hair = 100*Hfe.

Same problem, but the gap consists of 2 parallel spacers, one paper w/ mu_r = 1, the other cobalt alloy w/ mu_r = 10. Here the boundary condition is that the H field tangential components are equal for the 2 media. So we get Hpaper = Hco. But the B fields scale in direct relation to the relative permeabilities of the media, i.e. Bco = 10*Bpaper.

That's the long & short of it. Another example for magnetic fields is an inductor driven by a current source or voltage source plus series resistor. If 2 identical windings are placed in series w/ current source, the 1st being air cored, the 2nd being a ferrite w/ mu_r = 100, the H values are the same for both. The B values scale so that Bfe = 100*Bair.

If we place the windings in parallel, & excite them from an ac voltage source, the B values are the same. But the H values scale inversely w/ mu_r. Hence Hair = 10*Hfe.

Which one depends on the medium & which one is medium-independent can vary. Between E/D, & H/B, it can go either way. Also, B is magnetic flux density, whereas H is magnetic field intensity. Elecric flux density, aka electric displacement, is called D, whereas electric field intensity is E. I hope I've helped.

Claude

Last edited: Mar 17, 2011
6. Mar 18, 2011

### audioresearch

Re: what's the difference between magnetic flux density,B and magnectic field density

Here is an even more profound question.

If an H field is supposed to be due to current external to the material, then how can an H field exist inside a magnet that is floating in a vacuum with no external current source anywhere near it????

Shouldn't H be zero since there is no source of a magnetic field that is outside the magnet imposing its field upon the magnet?

I've seen textbooks that show an H field inside a magnet and I can't understand how that can be. I am not talking about when the magnet is being magnetized by an external H field and the well known B-H hysterisis curve applies, I'm talking about long after the magnetizing source has been shut off or moved well away from the magnet.

Last edited: Mar 18, 2011
7. Mar 18, 2011

### cabraham

Re: what's the difference between magnetic flux density,B and magnectic field density

H & B fields can & do exist in magnets. Research the following: paramagnetism, diamagnetism, ferrimagnetism, & super-paramagnetism. Let me know if clarification is needed.

Claue

8. Mar 19, 2011

### Stretto

Re: what's the difference between magnetic flux density,B and magnectic field density

D and H are fields that represent the changes due to the addition of material in a special way.

Suppose you look at an electric field with and without a material in it. You will notice that with a material there can be a change in the electric field. How? A material generally isn't a source of charge and generally doesn't have free currents in it.

This is then due to the bound charge in it(essentially the atoms and molecules). Hence it is due to the polarization of these charges since they can't are bound. If they could move then they would become free charge.

In many materials the change in the electric field is simply a change in it's magnitude. That is, E_mat is proportional to E_vac.

But we know that div(E) = p/e0. But p = p_f + p_p where p_f = free charge and p_p = polarization charge. p_p can only be due to multipoles of order > 1(else it would be part of p_f). Now p_p = div(P) and which we can rearrange the equations to get

div(e0*E + P) = p_f

We then say D = e0*E + P

So D is sort of the polarization field

We have the equation

div(D) = p_f which is analogous to div(E) = p

charge is a source of E but only free charge(as if we are dealing with a vacuum) is a source for D.

D is sort of the electric field(but not quite) as if the material were not there if. D = e0*E + P sort of takes E, which takes into account the material, and undoes what the material did. This is not completely true because D is not uniquely determined by div(D) = p_f.

The permittivity is a measure of how a material "condenses" the electric field in a material. When a material has polarization it "supports" this condensation because the multipoles arrange themselves so that the electric field outside do not have to travel as far. Think of a disk in in electric field. If the disk is simply polarized then it's as if one has a single dipole. Since we know that electric field lines terminate on charge the field lines will "head to" the sides of the disk. They do this because in some sense it is the path of least resistance. It's Hard to write in words without pictures but if you draw it you should see what I mean.

Now the electric field E will tell us this directly because charge is charge(it includes all the bound and free charge). That is, if we plot E we will see how the material affects it. Because the shorted path or the shorted field line between two charges with no other charge in between is a straight line between them, in a polarized material the shortest path is usually a straight line from one end of the material to the other(assuming a homogeneous material). That is, a polar material the multipoles tend to line themselves up and the electric field will take the shortest path from pole to pole. They tend to do this in linearly because it is the shortest path but not always the case tending on the material.

D basically gives us the difference between the electric field and the polarization field and more material independent as it the free electric flux density. It's not the electric field nor is it proportional to it. They have different units and generally just point in the same direction with similar magnitudes.

Now D is analogous to H(or B depending on how you look at it). Instead of free charge density source we have free vorticity source.

There is no correct way to separate them. We could have easily started with D and H and developed all of Electrodynamics. You do not even need to define D and H to have Electrodynamics. They are simply extra quantities that can help us in some cases when dealing with certain problems(Sometimes it is easier to think in terms of D and/or H).

The main thing to realize is that the difference between E and D and H and B have to do with materials. Make sure you understand polarization and magnetization first then attempt to see how these affect the E and B field then you should be able to arrive at how to interpret D and H for yourself. (I gave an outline for E and D above and B and H are analogous both conceptually and mathematically)

9. Mar 19, 2011

### audioresearch

Re: what's the difference between magnetic flux density,B and magnectic field density

Hi,

Thanks for your post.

Yes, I do need a further clarification regarding part of the definition of H that I've seen.

I've seen H defined in part as the external field imposed upon a material by an external source and that H field in turn induces a magnetization field inside the material M (both H and M are vectors) and the total magnetic field in the material is B = M + H (ignore any constants I left out, if any).

Is that always true? Is that always false? Is that true only in certain situations? If I'm using wrong definitions in whole or part, please point that out.

10. Mar 19, 2011

### Stretto

Re: what's the difference between magnetic flux density,B and magnectic field density

No, H and B are not simply related. They just happen to be "coincident" for most materials. You have to realize the units are totally different. In some materials the relationship between B and H can be quite complex.

You have to look at H in the analogous manner than you look at D. I could give my interpretation but it would follow exactly what I did with D except curls/vorticities would replace div/charge density and a slightly more complex notation that goes along with it.

In any case it doesn't matter too much in some sense they are all artificial. Ultimately you want to start thinking in terms of special theory of relativity where these concepts are less important. The potentials are generally more important than the fields.

http://en.wikipedia.org/wiki/Magnetic_field#H_and_B_inside_and_outside_of_magnetic_materials

Suppose you calculate H in the way they give. You have H, not B. To get B you must do additional work. It means you must know M or in some cases much more.

Now, since H = B/u0 - M and M is in the same direction, usually, as B, we can say in some sense that H is independent of the material(since that is represented by M).

Again, if you were to calculate H and D for a vacuum and for a material you would find they would be identical unless somehow the material was creating free charge/free current(which in most materials this doesn't happen).

If you calculated E and B for the two cases they would be different.

H and D can also be thought of as how E was defined from force. We take a "test" charge and compute the force on the test charge. E is the "force" if there were no charge at the point. H and D are the "fields" if no material is there. (but note in both cases the units are different)

11. Feb 13, 2012

### Bud Wood

Re: what's the difference between magnetic flux density,B and magnectic field density

The magnetic field strength (H) should not be confused with the magnetic flux density (B); the flux density is the field strength per area through which it is measured. The magnetic flux (Ф) is simply the total flux sans any reference to the area in which it exists.

12. Aug 9, 2012

### bck

Re: what's the difference between magnetic flux density,B and magnectic field density

Thanks for the answers, but it still doesn't clarify everything for me. Whenever I start to think I understand it, it is contradicted by another reply.

Now I have a question about a specific situation. Suppose I have a ferromagnetic material (thin elliptical layer), and I measure the behaviour of the magnetization (tunnel junction, so the behaviour is measured by measuring the resistance of the junction, this changes nothing about the magnetization). Anyway, I apply external magnetic fields by running currents through wires. The field-strength depends on the current, and I use the current to calculate the applied field. The field only depends on the current through the wire, not on the ferromagnet. Is the field I apply the B or H-field?

Last edited: Aug 9, 2012
13. Aug 10, 2012

### PhilDSP

Re: what's the difference between magnetic flux density,B and magnectic field density

The units and dimensions are different between B and H, and likewise between D and E. The fields are conceived (and measured) as force on a line element while the flux densities are conceived and measured as flux through an area.

The relationships B = μH and D = εE mentioned above hold true only in the SI system and in certain media. In the Gaussian system B = H and D = E in a vacuum.

14. Aug 10, 2012

### PhilDSP

Re: what's the difference between magnetic flux density,B and magnectic field density

That's your choice and may depend on whatever equation relating the current to the fields you wish to use. The current and charge density also are related to the fields by μ and ε (or the constant 1). E and B are usually used for bound charges while D and H are often used for free charges.

Last edited: Aug 10, 2012
15. Aug 10, 2012

### meldraft

Re: what's the difference between magnetic flux density,B and magnectic field density

The only real distinction mathematically is the magnitude. If, for instance, you plot Laplace's equation in a domain, the shape of the field lines will be exactly the same no matter the constant you use. The colormap (thus, the magnitude) of the field is the only thing that changes by using a different constant.

16. Aug 10, 2012

### bck

Re: what's the difference between magnetic flux density,B and magnectic field density

Thank you PhilDSP, I use CGS units, so, correct me if I'm wrong, in that case there is no difference at all between using B or H. Hence, it doesn't matter whether I plot the magnetization as a function of field in Gaussian or Oersted.

Now I remember why I favoured CGS!

17. Aug 10, 2012

### vanhees71

Re: what's the difference between magnetic flux density,B and magnectic field density

In the following I use rationalized Gaussian (Heaviside-Lorentz) units.

On a fundamental level there is only one electromagnetic field, which is described by the six components of an antisymmetric tensor in Minkowski space, $F_{\mu \nu}$. In an inertial frame it can be split in the time-space and space-space components and thus mapped to two three-vectors (more specific into a polar and a axial vector), $\vec{E}$ and $\vec{B}$. The fundamental field equations read
$$\epsilon^{\mu \nu \rho \sigma} \partial_{\nu} F_{\rho \sigma}=0, \quad \partial_{\mu} F^{\mu \nu}=\frac{1}{c} j^{\nu}.$$
Here $j^{\nu}$ is the total four-current of the electric charge. It takes explicitly into account all charges present.

For the case of macroscopic matter this set of equations cannot be solved exactly, and one uses approximations to simplify the task. Instead of treating the four-current of each atomic nucleus and electron as sources of the fields, one considers the situation where the wavelength of the electromagnetic field are large compared with the interatomic distances and coarse grains over spatial distances that are small compared with the considered wavelengths of the em. field but large compared with the atomic distances. In this way the matter is discribed in a continuum-mechanics like way. The four current then can be split into two pieces: One reflects the response of the charges constituting the atoms to the imposed electromagnetic field and the other takes into account explicitly brought in external charges and currents. The first part is, if the external electromagnetic fields are small compared to the interatomic fields, treatable in the linear-response approximation. This part of the currents after coarse-graining can be lumped over to the left-hand side of the equations in terms of induced polarization and magnetization of the matter. Then one introduces auxilliary fields $\vec{D}$ and $\vec{H}$. They can be put into fourdimensionally covariant form by putting them together into components of another antisymmetric four tensor $H_{\mu \nu}$. Then the Maxwell equations read
$$\partial_{\mu} \epsilon^{\mu \nu \rho \sigma} \partial_{\nu} F_{\rho \sigma}=0, \quad \partial_{\mu} H^{\mu \nu} = J^{\nu}.$$
Here $J^{\nu}$ is only that part the four current that is brought into to system as external charges and currents.

In addition one has in linear response approximation the consituent equations which give the linear mapping from $H_{\mu \nu}$ to $F_{\mu \nu}$ which in the most general form is a space-time integral with a tensor. In the most simple case of a homogeneous and isotropic medium and in the non-relativistic limit of the matter it's reduced to the simple forms with permitivity and permeability of the medium, $\vec{D}=\epsilon \vec{E}, \quad \vec{H}=\vec{B}/\mu$. The different way permitivity and permeability entering these equations is solely for historical reasons and stems from a time where these quantities were not understood in the above given simple terms as emergent phenomena of the one fundamental em. field in interaction with matter in a macroscopic effective theory of macroscopic matter of the underlying fundamental microscopic theory of elementary particles and fields that constitute the matter.

Another issue is the use of other units. The physics doesn't change only because we introduce other units to measure quantities. In the SI one has additional conversion factors to adapt the somewhat unhandy quantities for charges and currents to more convenient numbers for everyday use. That's why in electric engineering we measure charges not in units of elementary charges but in Coulombs and so on. The price we have to pay for this convenience is simply that the fundamental equations look a bit more complicated, but these has no significance physics wise. In the vacuum we still have only one electromagnetic field, represented by $\vec{E}$ and $\vec{B}$, but they are now measured in different units, and there appear formal permittivity- and permeability-looking-like conversion constants $\epsilon_0$ and $\mu_0$. They are arbitrarily chosen by the definition of the SI unit Ampere for electric currents, leading to $\mu_0:=4 \pi \cdot 10^{-7} \; \mathrm{N}/\mathrm{A}^2$. They are related to each other by $c=1/\sqrt{\epsilon_0 \mu_0}$, where $c$ is the speed of light in vacuum.

The speed of light in vacuum is nothing else than the fundamental "limiting speed" of Minkowski space and its value is thus also a simple arbitrary conversion factor to relate the unit of time (seconds) with the unit of lengths (metre) which again is to make everyday quantities to more convenient quantities than the use of "natural units", where $c=1$. In fact in the SI the speed of light is a defined exact value.

18. Aug 10, 2012

### DrDu

Re: what's the difference between magnetic flux density,B and magnectic field density

Even then the latter equations only hold for the Fourier components. The relation between E and D (and H and B) can still be non-local. Furthermore, when the system is not inversion symmetric, D may depend also on B and H on E.

19. Aug 10, 2012

### vanhees71

Re: what's the difference between magnetic flux density,B and magnectic field density

True! That's why I wrote "in the most simple cases" ;-).

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